Synopsis

Wall shear stress (WSS) is a hemodynamic parameter which
can be estimated from 4D flow MRI. The aim of this work was to advance the surface
inward normal computation for complex (i.e. cone-shaped) vessel geometries and
thus to improve the accuracy of wall shear stress estimates. We propose a Gauss
gradient field approach to adapt to complex vessel courses and evaluate our
method using synthetic flow data and selected patient data. Results show that
correct inward normal definition is crucial for reliable WSS estimates, in
particular in cases where complex vessel geometries are present.

Introduction

4D flow MRI allows the image-based estimation of quantitative wall shear stress (WSS) in vascular structures. WSS ($$$\vec{\tau}$$$) at any vessel wall point is defined as $$\vec{\tau}=2\eta\dot{\epsilon}\cdot\vec{n}$$ with $$$\vec{n}$$$ = vessel surface inward normal, $$$\eta$$$ = dynamic viscosity, and $$$\dot{\epsilon}$$$ = velocity deformation tensor. Furthermore, the oscillatory shear index (OSI) is defined as: $$OSI=\frac{1}{2}\left(1-\frac{|\int_{0}^{T}\vec{\tau}\cdot dt| }{\int_{0}^{T}|\vec{\tau}|\cdot dt}\right)$$ Previous works studied the impact of MRI acquisition parameters on WSS estimates1 though the burden of inaccurate inward normal definition is unknown. Most commonly, WSS and OSI are estimated by analyzing reformatted 2D planes2-4 and the inward normal at each wall point is only comprised of the vector component lying on the respective plane. However, in case of complex vessel geometries (e.g. stenosis, dilatation) the inward normal may deviate from the analysis plane.The aim of this work was to (1) introduce a method to accurately compute surface inward normals in cone-shaped vessel regions given the typical 4D flow MRI image data, and to (2) study the systematic error of WSS/OSI estimates when only the inplane inward normal component is considered.

WSS
computation. Vessels were segmented from the generated PC-MRA image using a watershed-based algorithm. For each case, one analysis plane was placed in the
dilated/stenotic vessel region and vessel contours were tracked through all
time frames (fig. 2). The binary vessel mask was used to derive a Gauss
gradient field ($$$\boldsymbol{G}$$$) to advance the computation of the vessel inward
normal at each sampled contour point $$$\boldsymbol{x}$$$:
$$\vec{n}_\boldsymbol{x}=\vec{a}_\boldsymbol{x}+\vec{g}_{\boldsymbol{x},p}$$ where $$$\vec{g}_{\boldsymbol{x},p}=\left(\boldsymbol{G}_\boldsymbol{x}\cdot\vec{p}\right)\vec{p}$$$
is the projection of $$$\boldsymbol{G}_\boldsymbol{x}$$$ (local Gauss
gradient vector) onto the unit plane normal $$$\vec{p}$$$, and $$$\vec{a}_\boldsymbol{x}$$$ is the
conventional inward normal lying on the analysis plane. WSS was then estimated
at each contour point using a B-spline image function based approach5 with η = 0.0032 Pa s. Regarding synthetic data, we analyzed single
timepoint WSS (WSSsingle); regarding in-vivo data, we analyzed WSS at
peak velocity (WSSpeak) and OSI. Estimates were binned into twelve
angular segments to compute mean±STD per segment. All image analysis was
implemented using the MevisLab6 framework.

Results

Both synthetic and in-vivo data results
show noticeably deviated inward normals when the Gauss gradient field is
incorporated (fig. 1, 2) with angle deviations ranging from 0.00-0.70 rad for
in-vivo data and 0.41-0.43 rad for the pipe phantom. With adapted inward normals,
synthetic data results show an increase in WSSsingle by 45.9±49.6% (fig.
3). Regarding in-vivo data results, we observe
differences in WSSpeak of
8.5±7.5%,
24.4±15.5%, 18.1±12.1%,
21.6±18.2%,
25.8±23.9%, and
10.9±9.7% (fig. 4), and in OSI of
15.8±13.3%,
10.5±10.8%,
23.1±23.2%,
7.2±8.0%,
4.6±6.0%, and
3.0±3.2% for cases 1-6, respectively (fig. 5). All
numbers indicated above are given as mean±STD over all twelve segments.

Discussion

Synthetic data results show a clear
underestimation when inward normal vectors are not adapted to the local vessel
geometry. Here, an outlier is present due to a very subtle flow profile and
thus comparably small WSSsingle values, which challenges the assessment in terms of
relative differences. Synthetic data should be optimized to better represent flow
velocities as given in the in-vivo data.
In-vivo data results clearly show that
adapted inward normal vectors result in strong deviations of WSSpeak and OSI, which
should not be neglected when using these parameters in a clinical study.
Particularly, differences are very pronounced for high angular deviations. Concering limitations, this study did not assess differences in
WSS/OSI contour-point-wise (only segment-wise) and thus did not correlate each contour
point WSS/OSI value with the angular deviation of the conventional and adapted
inward normal at this point.

Conclusion

The proposed method may be used to more precisely define vessel inward
normal vectors for increased WSS and OSI estimation. Particularly, this is
crucial where the vessel wall depicts a cone-shaped course.

Figures

Synthetic data representing steady
flow through a pipe with narrow. Left:
vector visualization (color coding = direction) of velocity field with
simulated flow from right to left and delineated vessel wall contour at the
region of interest (red). Right: Contour
point samples of the vessel wall contour (red dots) and inward normal vectors
defined by the conventional method (white arrows) and by integrating the gauss
gradient field of the vessel mask to adapt to the vessel course (red arrows).

Close-up
view of in-vivo data vessel surface
rendering (grey) with vessel wall inward normal vectors at contour points (red
dots). For each case, white arrows and red arrows represent the normal vectors
defined by the conventional method and by our approach, respectively. Angle
deviations between white and red vectors (computed at each contour point) are
given in median/min/max radian for each case.

Synthetic data WSS
estimates. Top:
Absolute differences of WSSsingle estimates comparing the
conventional (red) with the adapted (blue) inward normal vector computation.
Bottom: Relative differences of WSS estimates with the conventional method considered
a reference. Green bars depict a ±5% margin. Here, results show that adapted
inward normals effect an increase in WSS. Note that absolute WSS values are
very low compared to in-vivo results, which is due to low overall flow
velocities in the pipe phantom. This may also cause outliers as seen for the
first segment.

In-vivo
data WSSpeak estimates for six cases. Top:
Absolute differences of WSSpeak estimates comparing the conventional
(red) with the adapted (blue) inward normal vector computation. Bottom:
Relative differences of WSSpeak estimates with the conventional
method considered a reference. Green dashed lines depict a ±5% margin. Here,
results show that adapted inward normals effect an increase/decrease in WSSpeak
- depending on the specific case but also on the evaluated segment within one
particular case. This is mainly due to the inhomogeneous flow profile observed
in the in-vivo (patient) data.